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1. Field of the Invention
This invention relates to the method of comparing in three dimensions the steric and electrostatic fields exerted by molecules with similar binding affinities for a common molecule, and extracting by cross correlation of the fields, the most important common topological features related to the observed differences in binding affinities among those molecules. This method is particularly useful in understanding structure/function relationships in biological chemistry.
2. Description of Related Art
During the past three decades modern biology has come to recognize the importance of the three-dimensional conformation/shape of biological molecules in relation to the observed function and activity of these molecules. Beginning with the first identification of alpha helical structures in proteins through the solution to the structure of DNA as a hydrogen bonded intertwined double helix to current studies by X-ray crystallography of enzyme-substrate complexes, appreciation of the role of shape as a determining factor has continually increased. In fact, it is now understood that a proper description and understanding of the functioning of most biological macromolecules is dependent on an understanding of the three-dimensional shape of the molecules. The situation is often analogized to that of a three-dimensional jigsaw puzzle, where the parts which must fit together interlock in specific patterns in three dimensions. It is now realized that the binding of a molecular substrate to an enzyme is determined by the ability of the substrate to fit a notch/groove/cavity within the enzyme in such a manner that the substrate is both mechanically and chemically stabilized in the correct three-dimensional and thermodynamic orientation to promote the catalytic reaction. Similarly, it has long been recognized that the highly specific binding of antibodies to antigens is accomplished by the recognition by the antibody of the surface shape specific features of the antigen molecule.
Not only is the understanding of these three-dimensional puzzles important to a fundamental understanding of enzymology immunology, and biochemistry, but such studies are of major interest to therapeutic drug researchers. Most drug effects are accomplished by the binding of a drug to a target receptor molecule. To the extent that the nature of the binding is more fully understood, it should be possible to design drugs which
bind to their target molecules with greater precision and effect than even naturally occurring compounds. This therapeutic quest is especially important in cancer research where the generalized side effects of many therapeutic drugs are undesirable and more specific drug interactions are desired.
Along with the recognition of the importance of the three-dimensional stereo conformation of biomolecules has come an appreciation of just how difficult it is to understand how the conformation of the molecules is related to their activity. At the present time, the only known method for determining exactly the three-dimensional shape of any biomolecule is by means of X-ray crystallography. While the number of biomolecules which have had their structure successfully determined by crystallography is growing rapidly, the total number remains relatively small, and an even fewer number have been studied in crystal form in conjunction with their bound substrates or ligands. Of the few ligand-biomolecule combinations which have been successfully analyzed by X-ray crystallography, there is still the open question as to whether the complex exists in a different conformational combination in solution than it does in the crystallized form used for the study, although the evidence suggests that there is no major difference.
The study of the three-dimensional conformation/shape of molecules is thus seen to be one of the core questions in modern molecular biology and biophysics. With the possible exception of the introduction in the not too far distant future of coherent X-ray lasers which may make the three-dimensional imaging of biological macromolecules considerably easier, there have been no fundamental advances in the instrumental techniques available during the last several years. Nor have recent advances in protein sequence determination, either by direct sequencing of the proteins or by sequencing of the precursor DNA molecules, been of much help in elucidating the three-dimensional structures since it was discovered early on that, due to the highly folded protein structure, amino acid side chains from vastly different sections of a protein are involved in the conformation of the receptor or binding site. Similar considerations are true with respect to antibody formation. Only recently has a proposal been made towards understanding the initiation of alpha helixes, perhaps the simplest tertiary protein structure, based on a knowledge of the amino acid sequence in a protein. See Presta, L. G.; Rose, G. D. Science 1988, 240, 1632.
Recognizing the difficult and lengthy time involved in obtaining X-ray crystallographic structures of biomolecules, researchers have pursued alternate, though less exact, paths towards obtaining information on the stereochemical binding of molecules. One such approach, taken by experimental chemists, has been to apply an understanding of basic chemical principles to analyze the likely binding sites of substrates. By examining the chemical structures of various ligands known to bind to a given protein, and relying on an understanding of generalized chemical and stereochemical principles, chemists have made educated guesses concerning which parts of the substrate/ligand would most likely be involved in binding to the protein. Based on these educated guesses, new compounds have been synthesized incorporating predicted reactive sites. The binding affinities of the new substrates for the desired protein have been measured. Some reasonable measure of success in understanding stereochemical binding has been achieved by this empirical method, but failure has been much more frequent than success. This scheme, though rational, is basically one of trial and error and does not lead to a coherent approach to finding or designing new molecules with the desired binding affinities.
Attempts have been made over the years to place the understanding of stereochemical interactions of biomolecules and the development of new substrate molecules on a more quantitative footing. These approaches attempt to systematically relate differences in structures of similar substrate molecules to differences in their observed biological activities. Thus, a "structure activity relationship" (acronymed SAR) is sought for a given class of substrates/ligands. To the extent that these approaches have now been quantified, they are now referred to as "quantitative structure activity relationships" (acronymed QSAR). Generally, the relationship sought in formulating a QSAR is cast in the simplest possible format, that of a linear combination of elements. Thus the measured biological value, V, is sought to be explained by a series of terms, A, B, C, etc. as the linear combination: V=A+B+C+ . . . The QSAR approach can be used to relate many measures/properties of molecules which are somehow reflective of their structure, such as partition coefficients and molar refractivity. In the past these indirect measures of shape have been used in QSAR studies since using direct measures of shape proved conceptually and computationally difficult. As the art has progressed, and as the structural differences used in QSAR studies have become primarily molecular shape differences, the field of "three-dimensional quantitative structure activity relationships" (acronymed 3D-QSAR) has evolved.
The 3D-QSAR approach quantifies chosen shape parameters and tests to see if a correlation can be found between those parameters and a biological variable, typically binding affinity. It has turned out to be a very complex problem to model the interaction between a ligand and its receptor. The principal difficulty has been finding a quantitative way in which to express the simple concept of shape. As is often the case, what is visually obvious to the human eye and brain is complex to describe quantitatively or mathematically. While describing shape is difficult enough, searching for similarities in shape using shape descriptors which are, at best, inadequate turns out to be exceedingly difficult.
The general approach used in the QSAR methodology relies on the fact that, for most proteins, there are a number of chemical compounds or substrates having known structural differences which bind with differing affinities to the protein. The rationale behind the 3D-QSAR approach is that it should be possible to derive shape descriptors which, when applied to the various substrates, will reflect the different binding affinities. In 3D-QSAR a similar underlying assumption is made as in other QSAR approaches, i.e., that the relevant biological parameter, usually a binding affinity, can be represented as a linear combination of weighted contributions of the various shape descriptors for the substrate molecules. Once a whole series of substrates are described with the same shape descriptors, it should be possible to compare or correlate the shape descriptors and extract the critical shape determinants found to be associated with the differences in biological activity amongst the substrates.
From a knowledge of the most significant structural shape elements of the substrate or ligand, one could then infer the important elements of the receptor site on the protein. Ideally, in this process one would have at least as many substrates to compare as one had variables among the shape descriptors. Thus, a system of equations with the number of equations equaling the number of shape descriptors with unknown weighting coefficients would exist and could be solved exactly. However, in practice, it quickly became evident that, even with simplifying assumptions, using available shape descriptors to describe the properties of an unknown shape, the number of descriptor variables far exceeds the number of available substrates for which binding data is known. Thus, rather than getting an exact solution, it was found that approximating statistical methods had to be used to extract from the numerical shape descriptors the shape elements which best correlated with observed biological activity. However, until very recently statistical methods were not available which could extract useful information from a system of equations containing many more variables than equations.
During the past decade work has progressed in this field. From chemical analysis of substrate-protein complexes, it is known that the molecular interactions that produce an observed biological effect are usually non-covalent. Thus, the forces important for intermolecular association are believed to arise from hydrophobic, van der Waals (steric), hydrogen bonding, and electrostatic interactions. Attempts have been made to build shape descriptors based on these properties, but, unfortunately, the immense number of degrees of freedom and large labile protein-substrate complexes make the mathematical modeling of the shape of the complexes extremely difficult. Further simplifying criteria and assumptions were found to be necessary.
One such approach, entitled the Molecular Shape Approach developed independently by Simon, et al. (see Simon, Z; Badilenscu, I.; Racovitan, T. J. Theor. Biol. 1977, 66, 485 and Simon, Z.; Dragomir, N.; Planchithin, M. G.; Holban, S.; Glatt, H.; Kerek, F. Eur. J. Med. Chem. 1980, 15, 521) and Hopfinger, (see Hopfinger, A. J. J. Am. Chem. Soc. 1980, 102, 7196) compares net rather than location-dependent differences between molecules. That is, a shape characteristic of the total molecule is calculated in which the details of specific surface characteristics are merged into an overall molecular measure. The most active molecule in a series (in the sense of biological affinity) is considered to be a template molecule which has an optimal fit to the receptor site in the protein. Differences in activity amongst the series of substrate molecules are, therefore, potentially correlated by a multiple regression analysis with three structure (or shape) parameters definable for each member of the series. The shape parameters initially considered were either: 1) the common volume, 2) the volume possessed by the most active molecule, but not by the less active molecule, and 3) the volume possessed by the less active but not by the most active molecule in the series. Hopfinger describes these parameters as Common Overlap Steric Volumes and interprets them as quantitative measures of relative shape similarity.
More recently Hopfinger (see Hopfinger, A. J. J. Med. Chem. 1983, 26, 990) has constructed a new set of molecular shape descriptors derived from the potential energy field of a molecule. In this approach, Hopfinger uses molecular mechanics potentials as a means of estimating the molecular potential energy fields: ##EQU1##
In this equation the molecular potential energy field P.sub.u (R,.theta.,.phi.) at any given point (R,.theta.,.phi.) for molecule u is defined; a(T).sub.i and b(T).sub.i are the attractive and repulsive potential energy coefficients, respectively, of atom i of molecule u interacting with the test probe T which is treated as a single force center; Q.sub.i and Q(T) are, respectively, the charge densities of the ith atom and the test probe; .epsilon.(r.sub.i) is the dielectric term; n is the number of atoms in u, and (r.sub.i) is the distance between atom i and the test probe. Hopfinger suggests that pairwise field-difference [.DELTA.P.sub.u ] descriptors may correlate with biological parameters in a 3D-QSAR. Note, however, that this is a net molecular shape descriptor rather than a specific location-dependent shape descriptor.
A second approach is the Distance Geometry Method of Crippen. See for example Ghose, A.; Crippen, G. J. Med. Chem. 1985, 28, 333. In this approach the user must provide a "pharmacophore" or a list of potential receptor-binding atoms on each of the substrates/ligands having specified physicochemical properties. Knowledge of the pharmacophore comes from chemical studies of the binding properties of the given series of substrate molecules. The user must also provide a "binding site", a set of points in Cartesian space which are capable of interacting with a nearby pharmacophore atom, the magnitude of the attraction or repulsion depending on the nature of the atom. The geometrically allowed interactions between the ligand atoms and the binding site are determined. Each ligand is free to move or experience torsional deformations, in any fashion that minimizes the sum of its site points' energies of interaction with the "binding site". Thus, following Crippen, who again assumes a linear function for the interaction, the binding energy of a particular binding mode will be given by: ##EQU2## where E.sub.c is the energy of the conformation; C's are the coefficients to be determined by quadratic programming; i' is the type of site i; n.sub.s represents the number of site pockets; n.sub.p represents the number of parameters to correlate with that site pocket interaction; n.sub.o represents the number of atoms occupying that site pocket; P.sub.j represents the jth physiochemical parameter of the atom of type t.sub.k.
A successful 3D-QSAR is found when the sum of the energies of interaction obtained is suitably close to the binding affinities observed experimentally. The result provides both a receptor map (the position and nature of the "binding site" points) and, for each member of the series, an active conformation of that molecule. In both the Hopfinger and Crippen approach, it will be noted that an initial educated guess must be made for the choice of the active conformation of the molecule before the analysis can be done, and Crippen must further hypothesize an actual receptor site map in three dimensions.
Another major problem in any quantitative approach to shape analysis is the fact that, in solution, most compounds exist as a mixture of rapidly interequilibrating shapes or conformers. Generally, it is not even known which of the multiple conformations of a molecule is responsible for its measured biological affinity. Once again, educated guesses must be made to decide which of the many molecular conformations will be used in a 3D-QSAR analysis. The existence of multiple conformations further complicates the task of choosing the correct molecular orientation in which to make the comparison between the substrate molecules. Obviously, the ability of any shape measure to compare molecular shapes relies upon the correct relative orientation of the molecules when the shape measure is first determined. The same molecule when compared to itself rotated by 90.degree. would not likely show any common structural features. Therefore, several of the 3D-QSAR methods rely upon alignment rules to guarantee that only the variable or differing parts of the molecules make the greatest contribution to the shape comparison. It is obvious that the existence of multiple conformations for a given molecule complicates this task.
Typically then, a 3D-QSAR analysis starts out with many shape dependent parameters for a relatively few molecules whose biological activity, such as binding affinity, is known. This results in a series of linear relationships/equations relating the shape parameters to the biological measures having many more unknowns (columns) than relationships (rows). Except in the limiting cases of shape descriptors where oversimplifying assumptions are made, no statistical regression or correlation methods were available until recently which could give any possible hope of solving such a set of equations.